Optimal covariance change point localization in high dimensions

摘要

We study the problem of change point localization for covariance matrices in high dimensions. We assume that we observe a sequence of independent and centered p-dimensional sub-Gaussian random vectors whose covariance matrices are piecewise constant, and only change at unknown times. We are concerned with the localization task of estimating the positions of the change points. In our analysis, we allow for all the model parameters to change with the sample size n, including the dimension p, the minimal spacing between consecutive change points Delta, the maximal Orlicz-psi(2) norm B of the sample points and the magnitude kappa of the smallest distributional change, defined as the minimal operator norm of the difference between the covariance matrix at a change point and the covariance matrix at the previous time point.